# Questions tagged [sheaf-cohomology]

The sheaf-cohomology tag has no usage guidance.

318
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### Sheaf cohomology definition

I have seen multiple definitions for sheaf cohomology and wanted to ask for the reason.
One goes through injective resolutions and the other through flasque resolutions.
For paracompact bases it holds ...

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### Why does the associated sheaf vanish?

I am learning local cohomology from Hartshorne’s book Local Cohomology.
My question is about understanding a line in the proof of proposition 1.11 in this book.
The set-up for proposition 1.11 is that ...

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### Sheaf cohomology in number theory

I have read the first three chapters of Hartshorne and was wondering what are the applications of the notions presented in number theory or arithmetic geometry. I already know that the notion of ...

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### Using higher topos theory to study Cech cohomology

It seems to me that many classical statements about Cech cohomology should without much effort follow from Lurie's HTT, but I am struggling to fill out the details. I want to show that, given an ...

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### Singular cohomology to cohomology of quasi-coherent sheaf

Let $X$ be a projective nonsingular variety (integral Noetherian scheme of finite type that is proper over a field $k=\overline{k}$ such that $\Omega^1_X$ is locally free). Suppose one knows the ...

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### Unbounded acyclic resolutions

Let $\mathscr A$ be a Grothendieck abelian category. Then every object in $\operatorname{Ch}(\mathscr A)$ is quasi-isomorphic to a $K$-injective object [Stacks, Tag 079P]. In particular, for any left ...

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### Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper

In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...

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### Long exact sequence in Borel-Moore homology

The Wikipedia page for Borel-Moore homology states that for a locally compact set $X$ and a closed subset $Z$, if we write $U = X \setminus Z$ we have the following long exact sequence
$$\cdots \to H^{...

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### Homotopy invariant Bloch-Ogus cohomologies with a vanishing property

I am looking for examples (in any characteristic) of homotopy invariant Bloch-Ogus cohomology theories given by Zariski sheaves $\Gamma(n)$, such that $\Gamma(0) = \mathbb{Z}$ is the constant sheaf. ...

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### Surjection of a short exact sequence induced by spectral sequence (from paper of Schneider/Stuhler)

Let $K=\mathbb{Q}_p$ and $X$ a smooth separated rigid analytic variety over $K$ with coherent sheaf $\mathcal{F}$. Furthermore, $U \subset X$ is an open subvariety with admissible covering
$$ \ldots \...

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### Is $\mathbb{C}^*$ not irreducible, or is every locally constant sheaf on $\mathbb{C}^*$ constant?

I am running into contradiction from the following set of definitions, propositions, and assumptions. Can anyone spot where I'm off?
Definition A sheaf $\mathcal{F}$ on a topological space $X$ is ...

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### Formula for the Euler characteristic of a local system on $\mathbb{P}^1$

Let $X := \mathbb{P}^1$, $S\subset X$ a finite set of points, $U := X - S$, and $j : U\rightarrow X$ the inclusion.
Let $F$ be a complex local system on $U$ of rank $r$, and let $F_0$ be a typical ...

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### Isomorphism of cohomology bundles for smooth homotopic fibrations

Let $M,N$ be closed smooth manifolds. Let $f_0,f_1\colon M\to N$ be two smooth fibrations which are homotopic to each other in the class of smooth (equivalently, continuous) maps. Let $E^i_0, E^i_1$ ...

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### Zorn's lemma for Grothendieck sites

In every treatment of Grothendieck sites I can find, flasque sheaves are not defined in the way one would naïvely expect from ordinary sheaf cohomology; namely instead of saying that "restriction ...

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### Is the right adjoint to presheaf direct image interesting?

Let $X\overset{f}{\to}Y$
be a continuous map. It induces on presheaves a classical adjunction inverse image ⊣ direct image. However, the direct image functor has a further right adjoint, defined by ...

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### Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?

Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...

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### On the upper-bound for a type of quintuple Kloosterman sums

Sorry to disturb, dear experts here. I have a question involving the quintuple Kloosterman sum, and expect some hints to show the upper-bound.
My question is, for any $x,y,z,w,\delta \in \mathbb{Z}$ ...

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### Estimates for certain double-Kloosterman sums

Sorry to disturb. I encounter a double-Kloosterman sum, which needs some help from the experts here.
For any $q\in \mathbb{N}^+$, how can we estimate the type of sum
$$ \sideset{_{}^{}}{^{\ast}_{}}\...

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### A question involving the three-dimensional Kloosterman sum

Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here.
For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$,...

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### Cohomology of coherent sheaves on Deligne Mumford stacks

Suppose that $\cal X$ is tame Deligne Mumford stack with generic trivial inertia. Let $X$ be its muduli space and $f:{\cal X}\to X$ the projection.
Let $\cal F$ be a coherent sheaf on $\cal X$.
Is it ...

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### What's the point of fine sheaves? (As opposed to soft ones)

Why do people care to define fine sheaves? What useful property do they have for which softness is not sufficient?
some observations (because I feel guilty about a the one-line question):
The point ...

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### Deligne's integrality theorem in the setting of $ \mathbb{F}_{\ell}((t)) $-adic cohomology

Let $ \mathbb{F}_{q} $ be a finite field of characteristic $ p $ and $ \overline{\mathbb{F}_{q}} $ be an algebraic closure of $ \mathbb{F}_{q} $. Let $ X $ be a smooth projective variety over $ \...

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### Artin-Winters proof of semi-stable reduction theorem: details

I've been reading through Artin-Winters proof of the semi-stable reduction theorem (Degenerate fibers and stable reduction of curves) and found myself confused about the following detail—
Let $\...

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### Hypercovers with sieves

Consider a covering family $\{Y_i \to X\}$ and the induced sieve $R \subseteq X$, the subpresheaf of all maps to $X$ that factor through some $Y_i$. The family gives me an induced Cech nerve $C_\...

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### Is the cohomology $H^1(X, \mathcal{E}^\nabla)$ trivial, for the sheaf of constants of an algebraic connection $\nabla$?

Suppose that $\pi:X\to S$ is a flat morphism between Noetherian, integral schemes (of characteristic zero, if need be).
Let $\mathcal{E}$ be a locally free sheaf on $X$, and
$$\nabla:\mathcal{E} \to \...

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### Global section of pullback of an ideal sheaf

For a local ring $R$ with maximal ideal $\mathfrak{m}\subset R$ and residue field $\kappa$, and a flat morphism $f\colon X\rightarrow \mathrm{Spec} R$ of schemes, we consider the short exact sequence ...

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### Conditions for long exact sequence for line bundles on curve to degenerate?

Let $\varphi:X\to Y$ be a morphism of schemes of relative dimension 1, and $\mathcal{L}' \xrightarrow{g} \mathcal{L}$ an injection of line bundles on $X$.
The sequence
$$0\to \mathcal{L}' \xrightarrow{...

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### Type vs degree of a polarized abelian variety

Let $(A,L)$ be a polarized abelian variety. I know that the degree of the polarization is the Euler characteristic of $L$, so that
$d = \chi(L) = \dim H^0(A,L)$
since $L$ is ample.
I've read in a lot ...

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### Cohomology with coefficient in sheaf of morphisms of an algebraic group

Let $G$ be an affine algebraic group over ${\mathbb C}$. We denote the sheaf of morphisms from ${\mathbb A}^1$ to $G$ by $\bf G$. Then $H^1({\mathbb A}^1,\bf G)=0$ (Cech cohomlogy). Is this fact true? ...

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### Extensions for a short exact sequence on Grassmannians

$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\Ext{Ext}$Let us consider a $n$-dimensional complex vector space $V$ and denote by $G(k,n)$ the Grassmannian of $k$-planes in $V$. We use the ...

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### Base change of cohomology when the cohomology is a torsion

Let $(R,m,k)$ be a discrete valuation ring, where $k=R/m$. Let $X\rightarrow \mathrm{Spec}R$ be a projective, integral and flat $R$-scheme. Let $\mathscr F$ be a coherent sheaf such that $H^i(X,\...

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### Pushforward of structure sheaf along a torsor for a finite group

Let $\pi : P \to X$ be a torsor for a discrete, finite group $G$ of size $\#G = N$ on a scheme $X$. I want to compare $\pi_* \mathcal O_P$ with $\mathcal{O}_X$. Locally but not globally, $\pi_* \...

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### First cohomology of tangent sheaf of rational curve

Let $C$ be a reduced, connected, projective and purely one-dimensional scheme of finite type over a field $k$.
Suppose that $C$ is rational, i.e. that the normalisation of $C$ is a disjoint union of ...

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### Cech cohomology on product covers & Fréchet sheaves

My question is about the paper [Ka67].
Let $S, T$ be sheaves of nuclear Fréchet spaces over paracompact topological spaces $X, Y$, respectively; in particular, if $V \subset U$ are open subsets in $X$,...

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### Canonical coresolutions of cosheaves

Let $M$ be a sufficiently nice topological space (e.g. smooth manifold).
Recall that a (pre-)cosheaf in a category $\mathcal{C}$ over a topological space $M$ is essentially a $\mathcal{C}^\mathrm{op}$-...

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### First Chern class of torsion sheaves

Let $X$ be a smooth projective variety, $\mathscr T$ a torsion sheaf with irreducible support of codimension $1$, say $Z$. Then the first Chern class $c_1(\mathscr T)$ is of form $r[Z]$. Is there ...

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### Poincare polynomials for Borel Moore homology and fibrations

For an algebraic variety $X$ over $\mathbb{C}$, we denote $H_k(X)$ as its Borel-Moore homology of degree $k$. Let us define the Poincare polynomial associated it by
$$P(X)=\sum_{k\in \mathbb{N}}dim ...

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### understanding higher direct images of $\mathbb{G}_m$ for a finite Galois map

Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$, and let $\mu_r$ denote the group of $r$-th roots of unity, and moreover suppose $\mu_r$ (algebraically) acts on $X$ freely. Then $Y:= X/\...

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### $\bigoplus_{k=0}^{\infty}H^n(X,I^k\mathcal{F})$ is a finitely-generated $\bigoplus_{k=0}^nI^k-$graded module

Does anyone know where I can find a proof of the following result ?
Given a Noetherian ring $A$, a proper morphism of schemes $X\rightarrow \operatorname{Spec}A$, a coherent $O_X-$module $\mathcal{F}$ ...

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### Cohomology of divisors on Hirzebruch surfaces

Consider the Hirzebruch surface $\mathbb{F}_n = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(n))\rightarrow\mathbb{P}^1$. The Picard group of $\mathbb{F}_n$ is generated by ...

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### Topology on cohomology of a sheaf of topological groups

Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative topological groups on $X$. I am interested in the following question:
Is there a natural way to introduce topology on $H^i(X,...

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### The cohomology groups corresponding to a modified global sections functor

Let $\mathcal{F}$ be a sheaf on the big etale site of $Sm_k$. I am looking for a way to calculate a modified version of sheaf cohomology. Let $X$ be a smooth scheme and $Z$ a closed sub-scheme. After ...

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### Continuity of motivic cohomology under direct limit

Given the motivic complexes $\mathbb{Z}(n)$ on the big Zariski site of finite type smooth $k$-schemes denoted by $FinSm_k$, we pullback it to the smooth $k$-schemes i.e. $Sm_k$. For example for a ...

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### Künneth formula for local cohomology with support

In "Differential operators on the flag varieties" (http://www.numdam.org/article/AST_1981__87-88__43_0.pdf) by Brylinski, he uses on page 53 a Künneth formula for local cohomology with ...

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### Divisors whose restriction is big

Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X_A := f^{-1}(A)$.
...

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### Higher direct image with compact support of a constant sheaf

Let $f: X \to Y$ be a locally trivial fibration between locally compact spaces with fiber $F$. It is well known that for a constant sheaf $A_X$ on $X$, the higher direct images $R^n f_* A_X$ are ...

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### Global sections of multiples of a divisor

Let $D$ be an integral divisor on a smooth projective variety $X$. Consider the multiples $mD$ of $D$ for $m\geq 0$. Clearly, $h^0(X,mD) = 1$ for $m = 0$.
Is there any example where $h^0(X,mD) = 0$ ...

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### Extending $G$-torsors on open subsets of affine space

Let $k$ be a characteristic zero field, $V \subset \mathbb{A}^n_k$ an open subscheme, $G$ a split reductive group over $k$ and $T$ a $G$-torsor over $V$ (in the etale, equivalently fppf topology). ...

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### Cohomology of Grothendieck topology

My naïve cartoon picture of the construction of étale cohomology is this:
start with a scheme, associate to it a Grothendieck topology (making a site).
A functor from the Grothendieck topology to ...

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### Geometric meaning of coherent sheaves $\mathcal{F} \otimes \mathcal{O}_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$

Maybe it sounds like a silly question but I'm not able to figure out in my head the geometric meaning of "twisting" vector bundles (or more generaly coherent sheaf) over $\mathbb{P}^n$.
I ...